基於SVM的數字識別

KNN也能實現數字識別但需要保留所有的訓練樣本，支持向量機只需要保留支持向量就可以達到類似的效果

支持向量機本質上是一個二分類器

代碼如下：

# -*- coding: utf-8 -*-
#完整版的支持向量機 有核函數

from numpy import *
from time import sleep
#導入數據集
def loadDataSet(fileName):
    dataMat = []
    labelMat = []
    fr = open(fileName)
    for line in fr.readlines():#按行讀取
        lineArr = line.strip().split('\t')#對每行分割並剔除空格
        dataMat.append([float(lineArr[0]), float(lineArr[1])])
        labelMat.append(float(lineArr[2]))
    return dataMat,labelMat
#隨機選擇一個i!=j的數
def selectJrand(i,m):
    j=i #we want to select any J not equal to i
    while (j==i):
        j = int(random.uniform(0,m))
    return j
#數值太大太小時調整
def clipAlpha(aj,H,L):
    if aj > H: 
        aj = H
    if L > aj:
        aj = L
    return aj
#簡化的SMO算法
#輸入參數為（數據集，標簽集，常數C，容錯率，最大循環次數）
def smoSimple(dataMatIn, classLabels, C, toler, maxIter):
    dataMatrix = mat(dataMatIn);           #轉換為NumPy矩陣類型
    labelMat = mat(classLabels).transpose()#轉換為NumPy矩陣類型，並求轉置
    b = 0; 
    m,n = shape(dataMatrix)    #求矩陣的大小
    alphas = mat(zeros((m,1))) #生成一個0矩陣 列矩陣
    iter = 0                   #迭代次數
    while (iter < maxIter):
        alphaPairsChanged = 0  #用於記錄alpha是否已經優化
        for i in range(m):
            fXi = float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[i,:].T)) + b      #fXi是要預測的類別
            Ei = fXi - float(labelMat[i])#if checks if an example violates KKT conditions    #計算誤差
            if ((labelMat[i]*Ei < -toler) and (alphas[i] < C)) or ((labelMat[i]*Ei > toler) and (alphas[i] > 0)): #如果可以被優化
                j = selectJrand(i,m)#隨機選擇一個數
                fXj = float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[j,:].T)) + b  #fXj是要預測的類別 multiply表示各元素相乘,T是轉置
                Ej = fXj - float(labelMat[j])                                                #計算誤差
                alphaIold = alphas[i].copy()#python中的copy方法
                alphaJold = alphas[j].copy();
                if (labelMat[i] != labelMat[j]):
                    L = max(0, alphas[j] - alphas[i])
                    H = min(C, C + alphas[j] - alphas[i])
                else:
                    L = max(0, alphas[j] + alphas[i] - C)
                    H = min(C, alphas[j] + alphas[i])
                if L==H: 
                    print ("L==H"); continue
                eta = 2.0 * dataMatrix[i,:]*dataMatrix[j,:].T - dataMatrix[i,:]*dataMatrix[i,:].T - dataMatrix[j,:]*dataMatrix[j,:].T
                if eta >= 0: 
                    print ("eta>=0"); continue
                alphas[j] -= labelMat[j]*(Ei - Ej)/eta
                alphas[j] = clipAlpha(alphas[j],H,L)
                if (abs(alphas[j] - alphaJold) < 0.00001): 
                    print ("j not moving enough"); continue
                alphas[i] += labelMat[j]*labelMat[i]*(alphaJold - alphas[j])#更新i，與j的變化量相同但是方向相反
                #給兩個alpha值設置常數項b
                b1 = b - Ei- labelMat[i]*(alphas[i]-alphaIold)*dataMatrix[i,:]*dataMatrix[i,:].T - labelMat[j]*(alphas[j]-alphaJold)*dataMatrix[i,:]*dataMatrix[j,:].T
                b2 = b - Ej- labelMat[i]*(alphas[i]-alphaIold)*dataMatrix[i,:]*dataMatrix[j,:].T - labelMat[j]*(alphas[j]-alphaJold)*dataMatrix[j,:]*dataMatrix[j,:].T
                if (0 < alphas[i]) and (C > alphas[i]): 
                    b = b1
                elif (0 < alphas[j]) and (C > alphas[j]): 
                    b = b2
                else: 
                    b = (b1 + b2)/2.0
                alphaPairsChanged += 1
                print ("iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged))
        if (alphaPairsChanged == 0): 
            iter += 1 # alphaPairsChanged == 0 表示未更新
        else: 
            iter = 0  # alphaPairsChanged != 0 表示已更新
        print ("iteration number: %d" % iter)
    return b,alphas
#核轉換函數
#輸入參數為（）
#元組KTup給出了核函數的信息 元組的第一個參數描述核函數的類型
def kernelTrans(X, A, kTup): #calc the kernel or transform data to a higher dimensional space
    m,n = shape(X)
    K = mat(zeros((m,1)))
    if kTup[0]=='lin':   #線性核
        K = X * A.T   
    elif kTup[0]=='rbf': #徑向基核
        for j in range(m): #對矩陣的每個元素計算高斯函數的值
            deltaRow = X[j,:] - A
            K[j] = deltaRow*deltaRow.T
        K = exp(K/(-1*kTup[1]**2)) #divide in NumPy is element-wise not matrix like Matlab
    else: #遇到無法識別的元組，程序拋出異常
        raise NameError('Houston We Have a Problem That Kernel is not recognized')
    return K
#建立一個數據結構來保存重要值
class optStruct:
    def __init__(self,dataMatIn, classLabels, C, toler, kTup): #使用參數來初始化結構
        self.X = dataMatIn
        self.labelMat = classLabels
        self.C = C
        self.tol = toler
        self.m = shape(dataMatIn)[0]
        self.alphas = mat(zeros((self.m,1)))
        self.b = 0
        self.eCache = mat(zeros((self.m,2))) #第一列是標志位第二列是實際的E值
        self.K = mat(zeros((self.m,self.m)))
        for i in range(self.m):
            self.K[:,i] = kernelTrans(self.X, self.X[i,:], kTup)
#計算E值並返回，是從SMO中提取出來的        
def calcEk(oS, k):
    fXk = float(multiply(oS.alphas,oS.labelMat).T*oS.K[:,k] + oS.b)
    Ek = fXk - float(oS.labelMat[k])
    return Ek
#計算內循環的alpha        
def selectJ(i, oS, Ei):         #this is the second choice -heurstic, and calcs Ej
    maxK = -1; maxDeltaE = 0; Ej = 0
    oS.eCache[i] = [1,Ei]  #set valid #choose the alpha that gives the maximum delta E
    validEcacheList = nonzero(oS.eCache[:,0].A)[0]
    if (len(validEcacheList)) > 1:
        for k in validEcacheList:   #loop through valid Ecache values and find the one that maximizes delta E
            if k == i: continue #don't calc for i, waste of time
            Ek = calcEk(oS, k)
            deltaE = abs(Ei - Ek)
            if (deltaE > maxDeltaE):#改變最大的那個值
                maxK = k; maxDeltaE = deltaE; Ej = Ek
        return maxK, Ej
    else:   #in this case (first time around) we don't have any valid eCache values
        j = selectJrand(i, oS.m)
        Ej = calcEk(oS, j)
    return j, Ej
#更新
def updateEk(oS, k):#after any alpha has changed update the new value in the cache
    Ek = calcEk(oS, k)
    oS.eCache[k] = [1,Ek]
#與smoSimple類似但是有改進     
def innerL(i, oS):
    Ei = calcEk(oS, i)
    if ((oS.labelMat[i]*Ei < -oS.tol) and (oS.alphas[i] < oS.C)) or ((oS.labelMat[i]*Ei > oS.tol) and (oS.alphas[i] > 0)):
        j,Ej = selectJ(i, oS, Ei) #this has been changed from selectJrand
        alphaIold = oS.alphas[i].copy(); alphaJold = oS.alphas[j].copy();
        if (oS.labelMat[i] != oS.labelMat[j]):
            L = max(0, oS.alphas[j] - oS.alphas[i])
            H = min(oS.C, oS.C + oS.alphas[j] - oS.alphas[i])
        else:
            L = max(0, oS.alphas[j] + oS.alphas[i] - oS.C)
            H = min(oS.C, oS.alphas[j] + oS.alphas[i])
        if L==H: 
            print ("L==H"); return 0
        eta = 2.0 * oS.K[i,j] - oS.K[i,i] - oS.K[j,j] #changed for kernel
        if eta >= 0: 
            print ("eta>=0"); return 0
        oS.alphas[j] -= oS.labelMat[j]*(Ei - Ej)/eta
        oS.alphas[j] = clipAlpha(oS.alphas[j],H,L)
        updateEk(oS, j) #added this for the Ecache
        if (abs(oS.alphas[j] - alphaJold) < 0.00001): 
            print ("j not moving enough"); return 0
        oS.alphas[i] += oS.labelMat[j]*oS.labelMat[i]*(alphaJold - oS.alphas[j])#update i by the same amount as j
        updateEk(oS, i) #added this for the Ecache                    #the update is in the oppostie direction
        b1 = oS.b - Ei- oS.labelMat[i]*(oS.alphas[i]-alphaIold)*oS.K[i,i] - oS.labelMat[j]*(oS.alphas[j]-alphaJold)*oS.K[i,j]
        b2 = oS.b - Ej- oS.labelMat[i]*(oS.alphas[i]-alphaIold)*oS.K[i,j]- oS.labelMat[j]*(oS.alphas[j]-alphaJold)*oS.K[j,j]
        if (0 < oS.alphas[i]) and (oS.C > oS.alphas[i]): 
            oS.b = b1
        elif (0 < oS.alphas[j]) and (oS.C > oS.alphas[j]): 
            oS.b = b2
        else: 
            oS.b = (b1 + b2)/2.0
        return 1
    else: 
        return 0
#有核函數的完整版的SMO算法
#輸入參數為（數據集，標簽集，常數C，容錯率，最大循環次數，核函數）
def smoP(dataMatIn, classLabels, C, toler, maxIter,kTup=('lin', 0)):    #full Platt SMO
    oS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler, kTup)
    iter = 0
    entireSet = True
    alphaPairsChanged = 0
    while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
        alphaPairsChanged = 0
        if entireSet:   #go over all
            for i in range(oS.m):        
                alphaPairsChanged += innerL(i,oS)
                print ("fullSet, iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged))
            iter += 1
        else:#go over non-bound (railed) alphas
            nonBoundIs = nonzero((oS.alphas.A > 0) * (oS.alphas.A < C))[0]
            for i in nonBoundIs:
                alphaPairsChanged += innerL(i,oS)
                print ("non-bound, iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged))
            iter += 1
        if entireSet: 
            entireSet = False #toggle entire set loop
        elif (alphaPairsChanged == 0): 
            entireSet = True  
        print ("iteration number: %d" % iter)
    return oS.b,oS.alphas
#計算WS
def calcWs(alphas,dataArr,classLabels):
    X = mat(dataArr); labelMat = mat(classLabels).transpose()
    m,n = shape(X)
    w = zeros((n,1))
    for i in range(m):
        w += multiply(alphas[i]*labelMat[i],X[i,:].T)
    return w
#測試徑向基核函數
def testRbf(k1=1.3):
    dataArr,labelArr = loadDataSet('testSetRBF.txt')
    b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, ('rbf', k1)) #C=200 important
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    svInd=nonzero(alphas.A>0)[0]#得到大於零的alpha值 從而得到支持向量
    sVs=datMat[svInd] #get matrix of only support vectors
    labelSV = labelMat[svInd];
    print ("there are %d Support Vectors" % shape(sVs)[0])
    m,n = shape(datMat)
    errorCount = 0
    for i in range(m):#利用核函數分類
        kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1))
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr[i]): errorCount += 1
    print ("the training error rate is: %f" % (float(errorCount)/m))
    dataArr,labelArr = loadDataSet('testSetRBF2.txt')
    errorCount = 0
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    m,n = shape(datMat)
    for i in range(m):#利用核函數測試
        kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1))
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr[i]): errorCount += 1    
    print ("the test error rate is: %f" % (float(errorCount)/m))    

#********以下為使用核函數支持向量機做手寫識別分類***********#    
#轉換為向量
def img2vector(filename):
    returnVect = zeros((1,1024))
    fr = open(filename)
    for i in range(32):
        lineStr = fr.readline()
        for j in range(32):
            returnVect[0,32*i+j] = int(lineStr[j])
    return returnVect
#加載圖像
def loadImages(dirName):
    from os import listdir
    hwLabels = []
    trainingFileList = listdir(dirName)           #load the training set
    m = len(trainingFileList)
    trainingMat = zeros((m,1024))
    for i in range(m):
        fileNameStr = trainingFileList[i]
        fileStr = fileNameStr.split('.')[0]     #take off .txt
        classNumStr = int(fileStr.split('_')[0])
        if classNumStr == 9: #二分類
            hwLabels.append(-1)
        else: 
            hwLabels.append(1)
        trainingMat[i,:] = img2vector('%s/%s' % (dirName, fileNameStr))
    return trainingMat, hwLabels    
#和testRbf差不多也是一個測試函數
def testDigits(kTup=('rbf', 10)):#設置了默認的核函數
    dataArr,labelArr = loadImages('trainingDigits')
    b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, kTup)
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    svInd=nonzero(alphas.A>0)[0]
    sVs=datMat[svInd] 
    labelSV = labelMat[svInd];
    print ("there are %d Support Vectors" % shape(sVs)[0])
    m,n = shape(datMat)
    errorCount = 0
    for i in range(m):
        kernelEval = kernelTrans(sVs,datMat[i,:],kTup)
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr[i]): errorCount += 1
    print ("the training error rate is: %f" % (float(errorCount)/m))
    dataArr,labelArr = loadImages('testDigits')
    errorCount = 0
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    m,n = shape(datMat)
    for i in range(m):
        kernelEval = kernelTrans(sVs,datMat[i,:],kTup)
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr[i]): errorCount += 1    
    print ("the test error rate is: %f" % (float(errorCount)/m)) 

結果如下：

》import svm
》svm.testDigits(kTup=('rbf', 10))
there are 125 Support Vectors
the training error rate is: 0.000000
the test error rate is: 0.005376

除此外要有公式推導